Standard Deviation on a Calculator Help

Standard deviation represents the dispersion of any process. It can be calculated as the square root of mean of squares of differences of variate values from their mean. Let us see the methods to calculate standard deviation on a calculator


Methods to Calculate Standard Deviation ( S.d.)

`sigma` = calculation1

N
Where N is the total frequency ∑fi.

The square of standard deviation is called as variance.

I like to share this Formula to Calculate Standard Deviation with you all through my article.

Standard deviation Calculations:

The change of origin and the change of scale considerably reduces the labour in the calculation of standard deviation. The formulae for computation of `sigma`  are given below:

1)      Short-cut method

`sigma`= calculation2

2)      Step – deviation method

calculation3

Where di = xi – A and di’ =     (xi – A ) / h, being the assumed mean and h the equal class interval.

Proof: We know that calculation6

Therefore                  calculation7

calculation8

calculation9

`sigma` =    calculation10
Implications of Standard Deviation Methods Discussed

(i)               Quartile deviation = 2/3 ( Standard Deviation)

(ii)             Mean deviation = 4 / 5 ( Standard Deviation )

Cor. If m1,`sigma`1  be the mean and S.D. of sample size n1 and m2,`sigma`2  be those for sample of size n2, then the S.D. `sigma`of the combined sample of size n1 + n2 is given by

(n1 + n2)  = n1`sigma`12 + n2`sigma`22 + n1D12 + n2D22

Where Di = mi – m, m being mean of combined sample.

From 4, we have ns2 = n`sigma`2 + n ( x̅ - A ) 2 where n is the size of the sample.

i.e. sum of squares of deviations from A = n`sigma`2   + n ( x̅ - A ) 2

now lets apply this result to the first given sample taking A at m. Then, sum of squares of deviations of n1 items from m = n1 + n1 (m1 – m) 2.

Similarly for second given sample taking A at m. Then, sum of the squares of deviations of the n2 items from m = n2`sigma`12 + n2 (m2 – m) 2.

Adding both the samples at deviation n1 and n2 i.e. (n1+ n2) from m

= n1`sigma`12 + n2`sigma`22 + n1 (m1 – m) 2+ n2 (m2 – m) 2

(n1+n2) `sigma`2 = n1`sigma`12+ n2 `sigma`12 + n1 D12+ n2 D12

This result may be extend to combination of any number of samples, giving a result of the form ( ∑ ni ) `sigma` 2 = ∑ ( ni`sigma`2 i 2 ) +∑ (ni Di2).